2.2 RS model in string theory

UpdateJump To The Next Update Information There have been interesting developments in realizing the Randall–Sundrum brane-world model in string theory. A concrete example was found in type IIB supergravity [170Jump To The Next Citation Point]. The bosonic part of the 10-dimensional action is given by (see for example [355])
∫ { 2 2 } --1-- 10 √ ---- − 2ϕ [ 2] F-(1) --1-- ¯ -&tidle;F(5) SIIB = 2 κ2 d x − gs e ℛs + 4(∇ ϕ) − 2 − 2 ⋅ 3!G (3) ⋅G (3) − 4 ⋅ 5! 10 ∫ --1-- ϕ ¯ + 8iκ2 e C (4) ∧ G(3) ∧ G (3) + Sloc . (44 ) 10
Here gs denotes the string metric. We have also defined the combined three-flux, G(3) = F(3) − τH (3), where τ = C (0) + ie− ϕ, and
&tidle;F(5) = F (5) − 1C (2) ∧ H(3) + 1B (2) ∧ F (3) . (45 ) 2 2
The term Sloc is the action of localized objects, such as branes.

There is a 10-dimensional solution similar to the one in the RS model

ds2 = h(y)− 12ημνdxμdx ν + h(y)12&tidle;gmndymdyn, (46 ) 10
where μ x are four-dimensional coordinates and m y are coordinates on the 6-dimensional compact manifold ℳ6. If there are N coincident D3-branes on the manifold, the 10-dimensional spacetime is described by AdS5 × X5 where X5 is a five-dimensional Einstein manifold. The warp factor is given by [418]
( )4 h(r) ∼ L- , (47 ) r
where r is the distance from the D3 branes in the &tidle;g mn metric and L is given by
′2 L = 4πagsN α . (48 )
Here gs is the string coupling constant, α ′ is related to the string mass scale ms as α′ = 1∕m2 s and a depends on X5.

Near the D3 branes, the geometry is AdS5 × S5. At large r, the product structure of the 10-dimensional metric breaks down and r ceases to be a good coordinate, and the 6D extra-dimensional space ℳ6 is not a product of a five sphere and one-dimensional spacetime. This gives a minimum distance r0. This cut-off acts as a TeV brane in the RS model. On the other hand, for r > rmax, the AdS geometry smoothly glues into a Calabi–Yau compactification. Thus, the compact manifold plays the role of the Planck brane in the RS model, acting as a cut-off of the spacetime. The minimum warp factor − 1∕2 h (r0) is determined by the flux associated with F3 and H3 and thanks to the warping, it is possible to realize a large hierarchy [170]

-h(r0)--≫ 1. (49 ) h(rmax)


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